If $\alpha > \beta > 0$ are the roots of the equation $a x ^ { 2 } + b x + 1 = 0$, and $\lim _ { x \rightarrow \frac { 1 } { \alpha } } \left( \frac { 1 - \cos \left( x ^ { 2 } + b x + a \right) } { 2 ( 1 - \alpha x ) ^ { 2 } } \right) ^ { \frac { 1 } { 2 } } = \frac { 1 } { k } \left( \frac { 1 } { \beta } - \frac { 1 } { \alpha } \right)$, then $k$ is equal to\\
(1) $2 \beta$\\
(2) $\alpha$\\
(3) $2 \alpha$\\
(4) $\beta$